On a damped Szego equation (with an appendix in collaboration with Christian Klein)

TL;DR

This paper studies how damping the lowest Fourier mode affects the cubic Szeg{"o} equation, revealing that certain initial conditions lead to unbounded Sobolev norms and providing a detailed analysis on a reduced phase space.

Contribution

It offers a new understanding of the damped cubic Szeg{"o} equation's dynamics, including a complete analysis on a reduced phase space and numerical support for broader initial data.

Findings

Existence of initial data with Sobolev norms tending to infinity

Complete phase space analysis on a 6-dimensional reduced space

Numerical simulations supporting the theoretical results

Abstract

We investigate how damping the lowest Fourier mode modifies the dynamics of the cubic Szeg{\"o} equation. We show that there is a nonempty open subset of initial data generating trajec-tories with high Sobolev norms tending to infinity. In addition, we give a complete picture of this phenomenon on a reduced phase space of dimension 6. An appendix is devoted to numerical simulations supporting the generalisation of this picture to more general initial data.